174 Stock Analysis Project
2023-03-01
Comparing Amazon, Microsoft, Nvidia, and Google stocks
174 Project
By: Jae Yun
Introduction:
The stock market is one of the most fascinating fields, as many stocks within the same industry follow the same price pattern. This is especially true in the technology industry where companies like as Amazon, Microsoft, Nvidia, and Google have become dominant companies, as they hold some of the largest market caps in the world. These companies have all experienced significant growth over the past decade, and the technology has outgrown all other industries and became the leading sector in the past few decades.
It is interesting to study these stocks in the industry because of the fact that we can determine the entire technology industry using just these four stocks. However, the main idea is that we can to make forecasts using the stocks and determine if each of the stocks are related to their technological counterparts, as it is a crucial component of a stock market analyst in predicting the future of the stock. While analyzing the technology industry as a whole can provide valuable insights into overall trends and economic conditions, predicting the performance of individual stocks can help investors and analysts make more informed decisions about their specific investments, especially ones as stable as Amzn, Msft, Nvda, and Googl.
While many companies prefer to diversify their stock portfolio, some companies tend invest in individual stocks from a certain industry as a way to generate higher returns, especially if they have the knowledge about the certain company they are investing in. This strategy is a high-reward type situation that would help the company thrive but also pose high risks to due the volatility of individual stocks and their strong reactions from current market trends. However, the time period from 2010 to 2019 was a relatively stable growth period in terms of political and economical stability, and thus, companies would have chosen a riskier approach during this decade.
Thus, we play the role of a data analyst in a company following a riskier investing scheme, with goal of utilizing each individual stock to predict their future performance and see which of the four stocks have a higher return of investment within the next year between the beginning of 2019 to end of 2019 based on the data from 2010 to the end of 2018. We want to see which of the four stocks will have the highest predicted performance of the 2019 year, in hopes that we can use the same model to predict the 2020 year.
For our project, we will use four weekly prices of Amazon, Microsoft, NVIDIA, and Google from the 10 year time frame between January 1st, 2010, to January 1st, 2020, and compare the datasets to each other. Our datasets were retrieved from Yahoo Finance, with 522 observations representing each week’s price opening for each of the four tech stocks.
We wanted to find a certain time frame in which a major event did not occur, so that we have a more reliable prediction that is based on the trend and seasonality of our data throughout the years, and rather not based on certain events, such as the coronavirus pandemic, the Russo-Ukrainian war, or the 2008 recession. Albeit the time frame from 2010 to 2020 had some events. This is a biased dataset because we have knowledge of the past and the events that have occured, but we want to consider our model forecasts if we hypothetically did not have any major stock-changing events since we want a more technical analysis/forecasts of the stock rather than having to consider actual events, which we can’t really quantify in our time series predictions.
FORMULAS USED:
Augmented Dickey-Fuller (ADF Test):
\[\Delta y_t = \alpha + \beta t + \gamma y_{t-1} + \delta_1 \Delta y_{t-1} + \cdots + \delta_{p-1} \Delta y_{t-p+1} + \varepsilon_t\]
where \(\Delta y_t\) is the first difference of the time series at time \(t\), \(\alpha\) is a constant term, \(\beta\) is the coefficient of a linear time trend, and \(\gamma\) is the coefficient of the lagged value \(y_{t-1}\). The terms \(\delta_i\) are the coefficients of the lagged first differences, up to a maximum lag of \(p-1\). The error term \(\varepsilon_t\) is assumed to be white noise.
Decomposition: Y(t) = T(t) + S(t) + R(t) where:
*Y(t) is the observed value of the time series at time t = T(t) + S(t) + R(t)
*T(t) is the trend component of the time series at time t \(T_t = f(t)\)
S(t) is the seasonal component of the time series at time \(S_t = \sum_{i=1}^n s_i\)
R(t) is the residual or random component of the time series at time t \(R_t = Y_t - T_t - S_t\)
ACF:
\[\text{ACF}(k) = \frac{\sum_{t=k+1}^n (Y_t - \bar{Y})(Y_{t-k} - \bar{Y})}{\sum_{t=1}^n (Y_t - \bar{Y})^2}\]
where \(k\) is the lag, \(n\) is the number of observations in the time series, \(Y_t\) is the observed value at time \(t\), and \(\bar{Y}\) is the sample mean of the time series.
PACF:
\[\rho_0 = 1\]
\[\rho_k = \frac{\text{cov}(Y_t, Y_{t-k})}{\text{var}(Y_t)} - \sum_{j=1}^{k-1} \phi_j \rho_{k-j}\]
where \(\rho_k\) is the partial autocorrelation at lag \(k\), \(\text{cov}(Y_t, Y_{t-k})\) is the covariance between \(Y_t\) and \(Y_{t-k}\), \(\text{var}(Y_t)\) is the variance of \(Y_t\), \(\phi_j\) is the estimated coefficient of the autoregressive (AR) model at lag \(j\), and \(k\) is the maximum lag for which the PACF is computed.
Differencing:
\(y't = y_t - y{t-52}\) For our 52 weekly period differencing
\(y't = y_t - y{t-1}\) For taking the first difference
where \(y_t\) is the observed value of the time series at time \(t\), and \(y'_{t}\) is the differenced value of the time series at time \(t\).
Sarima Model:
\[(1 - \phi_1 B - \cdots - \phi_p B^p)(1 - \Phi_1 B^s - \cdots - \Phi_P B^{sP})^d (y_t - \mu)\] \[= (1 + \theta_1 B + \cdots + \theta_q B^q)(1 + \Theta_1 B^s + \cdots + \Theta_Q B^{sQ})\varepsilon_t\] where \(y_t\) is the observed value of the time series at time \(t\), \(\mu\) is the mean of the time series, \(\varepsilon_t\) is the error term at time \(t\), and \(B\) is the backshift operator. The parameters \(p\), \(d\), and \(q\) correspond to the order of the autoregressive (AR), integrated (I), and moving average (MA) components, respectively. The parameters \(P\), \(D\), and \(Q\) correspond to the order of the seasonal AR, seasonal I, and seasonal MA components, respectively. The parameter \(s\) represents the length of the seasonal cycle.
The \(\phi\) and \(\theta\) parameters are the coefficients of the AR and MA terms, respectively, while the \(\Phi\) and \(\Theta\) parameters are the coefficients of the seasonal AR and seasonal MA terms
Ljung-Box Test
\[Q = n(n+2) \sum_{k=1}^h \frac{\hat{\rho}_k^2}{n-k}\]
where \(Q\) is the test statistic, \(n\) is the sample size, \(h\) is the number of lags being tested, and \(\hat{\rho}_k\) is the sample autocorrelation at lag \(k\). The null hypothesis of the test is that the autocorrelations up to lag \(h\) are equal to zero, indicating that the time series is a white noise process. The alternative hypothesis is that the autocorrelations are not equal to zero, indicating the presence of serial correlation in the data. The test statistic \(Q\) follows a chi-squared distribution with \(h-p\) degrees of freedom, where \(p\) is the number of parameters estimated in the time series model.
Box-Pierce Test
\[Q^* = n \sum_{k=1}^h \hat{\rho}_k^2\]
where \(Q^*\) is the test statistic, \(n\) is the sample size, \(h\) is the number of lags being tested, and \(\hat{\rho}_k\) is the sample autocorrelation at lag \(k\). The null hypothesis of the test is that the autocorrelations up to lag \(h\) are equal to zero, indicating that the time series is a white noise process. The alternative hypothesis is that the autocorrelations are not equal to zero, indicating the presence of serial correlation in the data.
The test statistic \(Q^*\) follows an approximately chi-squared distribution with \(h-p\) degrees of freedom, where \(p\) is the number of parameters estimated in the time series model.
Data Analysis:
The 5 V’s of our Data: Volume: We have a significant amount of data, ranging from weekly data for a span of 10 years in the stock market. Thus, we can use this vast amount of data to make our predictions more effectively Velocity: Our velocity of our data is the weekly aspect Variety: Albeit having 4 of some of the largest stocks in the technology industry, we have a variety in the sense that these prices are all mutually exclusive to their own stock, and each company has different behaviors. Veracity: Since we are working with data retrieved from Yahoo Finance, we have pinpointed accuracy in our data because these prices were reflected on the actual stock market prices during these times. We have a very strong accuracy that our data is true. Value: By analyzing this data, we can determine trends and make analysis of the data as well as look into the future of how the data will perform, and also use this forecasting to predict other aspects of the stock market or economy, since these powerhouse stocks are highly impactful of the tech industry.
Now, we can begin importing our data from yahoo, read csv, and plot the time series data:
Print summary of NVDA stock as reference
print(summary(NVDA_Weekly))## Date Open High Low
## Min. :2010-01-01 Min. : 2.265 Min. : 2.365 Min. : 2.163
## 1st Qu.:2012-06-30 1st Qu.: 3.720 1st Qu.: 3.850 1st Qu.: 3.576
## Median :2014-12-29 Median : 5.155 Median : 5.268 Median : 4.999
## Mean :2014-12-29 Mean :17.746 Mean :18.425 Mean :17.070
## 3rd Qu.:2017-06-28 3rd Qu.:35.491 3rd Qu.:36.924 3rd Qu.:33.289
## Max. :2019-12-27 Max. :69.573 Max. :73.190 Max. :67.900
## Close Adj Close Volume
## Min. : 2.240 Min. : 2.055 Min. :4.140e+07
## 1st Qu.: 3.693 1st Qu.: 3.416 1st Qu.:1.517e+08
## Median : 5.152 Median : 4.949 Median :2.195e+08
## Mean :17.828 Mean :17.529 Mean :2.516e+08
## 3rd Qu.:35.921 3rd Qu.:35.609 3rd Qu.:3.127e+08
## Max. :69.823 Max. :69.230 Max. :1.317e+09Check the class of our dataset.
class(NVDA_Weekly)## [1] "spec_tbl_df" "tbl_df" "tbl" "data.frame"As shown above, we have a tibble dataframe of our stocks we read in from our 4 csv files of the NVDA, GOOGL, MSFT, and AMZN stocks, with column values corresponding to the dates price at stock open, close, as well as the low and high prices during open, and finally, the volume. Our dataset ranges from the beginning 2010 to the end of 2019, calculating the weekly open prices for each week.
GOOGL_Weekly_Open_ts <- ts(GOOGL_Weekly$Open, start = c(2010, 1), frequency = 52)
NVDA_Weekly_Open_ts <- ts(NVDA_Weekly$Open, start = c(2010, 1), frequency = 52)
MSFT_Weekly_Open_ts <- ts(MSFT_Weekly$Open, start = c(2010, 1), frequency = 52)
AMZN_Weekly_Open_ts <- ts(AMZN_Weekly$Open, start = c(2010, 1), frequency = 52)
Our tech stocks have in common a steady increase weekly from 2010 to 2020.
Checking for NA values in our dataset:
## googl_isna nvda_isna msft_isna amzn_isna
## 1 0 0 0 0Constructing histograms:
As we see from our histograms, we have highly skewed data and requiring a transformation to normalize.
Use the Augmented Dickey-Fuller Test to check for stationarity in each stock
## GOOGL_ADF_P_VALUE NVDA_ADF_P_VALUE MSFT_ADF_P_VALUE AMZN_ADF_P_VALUE
## 1 0.1877035 0.656367 0.8858777 0.6416679Using our adf tests, we can tell that all of our stocks are non-stationary processes, since they are above the significance value (p value > 0.05) and thus we fail to reject the non-stationarity null hypothesis.
Transforming our Data to a Stationary Process
We will transform our dataset to remove our trend and seasonal components by performing a decomposition and extracting its residuals. First, we will get the first difference of our datasets because our data is non-stationary, and we need to remove our trends and seasonality. As we can see from our decomposition graphs, we clearly have a trend and seasonal aspect of our dataset:
# Google's Decomposition plot
plot(decompose(gtr))
# Nvidia's Decomposition plot
plot(decompose(ntr))
# Microsoft's Decomposition plot
plot(decompose(mtr))
# Amazon's Decomposition plot
plot(decompose(atr))
We can see that we have a clear trend and seasonality from our decomposition. Our random also has a slight bit of pattern towards the end. This signals that we can difference further to stabilize the variance and remove further trends/seasonalities.
We must omit the NA values that were generated from the decomposition function. Since we have seasonality at a period of each year, we then find the loss at lag = 52 since we have 52 weeks in a year to see if we can remove our seasonality component. If not, we can test our arima vs sarima model and see if we have a lower AIC if we add in the seasonal component to our arima.
googl_residuals <- na.omit(decompose(gtr)$random) %>% diff(lag = 52)
nvda_residuals <- na.omit(decompose(ntr)$random) %>% diff(lag = 52)
msft_residuals <- na.omit(decompose(mtr)$random) %>% diff(lag = 52)
amzn_residuals <- na.omit(decompose(atr)$random) %>% diff(lag = 52)Lets check our plots to see if there is a trend/stationarity in our residuals:
It seems our plots show some trends and seasonality. We can check other
factors as well.
ACF plots:
par(mfrow = c(2,2))
acf(googl_residuals, lag.max = 100)
acf(nvda_residuals, lag.max = 100)
acf(msft_residuals, lag.max = 100)
acf(amzn_residuals, lag.max = 100)
Our acf plot suggest that we have non-stationarity since our values are always passing the significance blue dashed line level and not gradually decreasing to zero as lag increases. We can also check our pacf plot:
par(mfrow = c(2,2))
pacf(googl_residuals, lag.max = 100)
pacf(nvda_residuals, lag.max = 100)
pacf(msft_residuals, lag.max = 100)
pacf(amzn_residuals, lag.max = 100)
We have decaying pacf plots which is good for stationarity, but we need to be sure we have stationarity by differencing even further because the ACF’s indicate non-stationarity.
Let’s check our differenced plot to see if there are any trends or seasonality:
Looking at our first difference, we see a significant improvement in having a constant mean compared to our non-differenced residuals, in addition, our plot does not have any trend or seasonality appearance
We can check our ACF plots:
par(mfrow = c(2,2))
acf(googl_residuals_diff, lag.max = 100)
acf(nvda_residuals_diff, lag.max = 100)
acf(msft_residuals_diff, lag.max = 100)
acf(amzn_residuals_diff, lag.max = 100)
We can see our acf plots are gradually converging to zero as lag increases, suggesting some sort of stationarity processes in our datasets, but could also suggest a hint of seasonality since there are spikes at the periods.
Running PACF tests
par(mfrow = c(2,2))
pacf(googl_residuals_diff, lag.max = 100)
pacf(nvda_residuals_diff, lag.max = 100)
pacf(msft_residuals_diff, lag.max = 100)
pacf(amzn_residuals_diff, lag.max = 100)
Our PACF plots aren’t decaying, and we see a small bit of seasonality.
We can check our ADF and KPSS tests to double check for stationarity,
and run our Arima model with a higher order of autoregression to account
for the pacf plot.
We run our Augmented Dickey Fuller test on each of our differenced residuals
data.frame(GOOGL_ADF_P_VALUE = adf.test(googl_residuals_diff)$p.value,
NVIDIA_ADF_P_VALUE = adf.test(nvda_residuals_diff)$p.value,
MSFT_ADF_P_VALUE = adf.test(msft_residuals_diff)$p.value,
AMZN_ADF_P_VALUE = adf.test(amzn_residuals_diff)$p.value)## GOOGL_ADF_P_VALUE NVIDIA_ADF_P_VALUE MSFT_ADF_P_VALUE AMZN_ADF_P_VALUE
## 1 0.01 0.01 0.01 0.01The ADF test suggests that our datasets are all stationarity. We can make sure we do not have other types of non-stationarity by running the Kwiatkowski–Phillips–Schmidt–Shin (KPSS) test:
data.frame(GOOGL_kpss_P_VALUE = kpss.test(googl_residuals_diff)$p.value,
NVIDIA_kpss_P_VALUE = kpss.test(nvda_residuals_diff)$p.value,
MSFT_kpss_P_VALUE = kpss.test(msft_residuals_diff)$p.value,
AMZN_kpss_P_VALUE = kpss.test(amzn_residuals_diff)$p.value)## GOOGL_kpss_P_VALUE NVIDIA_kpss_P_VALUE MSFT_kpss_P_VALUE AMZN_kpss_P_VALUE
## 1 0.1 0.1 0.1 0.1Since our datsets pass both our ADF and KPSS tests for stationarity, we can fit our arima time series models on our datasets.
Fitting our Arima models:
For our model, we will use the arima model without the seasonal component to see which models perform best. Our ranking metri will be using aic.
GOOGLE MODELS:
For our google residuals diff parameters, where we have it differenced, we don’t need to difference it further, and so d = 0. For our seasonal component, we can set P as 1.0 or 2.0 as they are significant at those seasonal periods as we see from our PACF graph. For Q, we would set it as 1 since we have a significant value at the 1.0 from the ACF graph. D = 0 since we do not need to get the seasonal difference. For q, we see a lot of values , where q can equal to 0, 1, 3, 10,… and p can equal 1, 3, 4, 6, 9,… and more, but to avoid overfitting our dataset and for computational sake, we can use the first significant value where p = 1 or q = 0 or 1 , as these values also gives us the best AIC.
Therefore, here are our possibilities for our arima dataset for Google: Sarima(p = 1 or 2, d = 0, q = 0 or 1, P = 1 or 2, D = 0, Q = 1)
After fine tuning our parameters, we ended up with the three best models: Arima(1, 0, 1)(1, 0, 1), Arima(1, 0, 0)(1, 0, 1), and Arima(2, 0, 1)(1, 0, 1)
## googl_model1.aic googl_model2.aic googl_model3.aic
## 1 1112.548 1114.544 1116.071These are our best models using the arima and sarima packages. Note that two of the parameters are the same, but are from different packages and thus can be considered as different models.
We will now see what best fits our other datasets:
NVIDIA MODELS:
For our nvidia residuals diff parameters, where we have it differenced, we don’t need to difference it further, and so d = 0. For our seasonal component, we can set P as 1.0 as they are significant at those seasonal periods as we see from our PACF graph. For Q, we would set it as 1 since we have a significant value at the 1.0 from the ACF graph. D = 0 since we do not need to get the seasonal difference. For p, we see a lot of values, where p can equal to 8, 9, 13,.. and q can equal 0, 8, 9,… and more, but to avoid overfitting our dataset and for computational sake we will consider the first three values of p and q
Therefore, here are our possibilities for our arima dataset for Nvidia: Sarima(p = 8 or 9 d = 0, q = 0, 7, or 8 P = 1, D = 0, Q = 1)
After fine tuning our data to get the best 3 AIC values, we ended up with the three best models: ARIMA(8, 0, 8)(1, 0, 1), AND ARIMA(9, 0, 8)(1, 0, 1), ARIMA(8, 0, 7)(1, 0, 1)
## nvda_model1.aic nvda_model2.aic nvda_model3.aic
## 1 1069.107 1103.472 1077.038MICROSOFT MODELS:
For our msft residuals diff parameters, where we have it differenced, we don’t need to difference it further, and so d = 0. For our seasonal component, we can set P as 1.0 or 2.0 as they are significant at those seasonal periods as we see from our PACF graph. For Q, we would set it as 1 since we have a significant value at the 1.0 from the ACF graph. D = 0 since we do not need to get the seasonal difference. For p, we see a lot of values, where p can equal to 1, 12, 14,.. and q can equal 0, 1, 6, 17,… and more, but to avoid overfitting our dataset and for computational sake, we will consider the first three values of p and q.
Therefore, here are our possibilities for our arima dataset for MICROSOFT: Sarima(p = 1, 12, 14 d = 0, q = 0, 1, 6, 17 P = 1.0 or 2.0, D = 0, Q = 1)
After fine tuning our data to get the best 3 AIC values, we ended up with the three best models: ARIMA(12, 0, 6)(1, 0, 1), ARIMA(12, 0, 0)(1, 0, 1), ARIMA(12, 0, 1)(1, 0, 1)
## msft_model1.aic msft_model2.aic msft_model3.aic
## 1 1340.611 1335.333 1337.569AMAZON MODELS:
For our amazn residuals diff parameters, where we have it differenced, we don’t need to difference it further, and so d = 0. For our seasonal component, we can set P as 1.0 or 2.0 as they are significant at those seasonal periods as we see from our PACF graph. For Q, we would set it as 1 since we have a significant value at the 1.0 from the ACF graph. D = 0 since we do not need to get the seasonal difference. For p, we see a lot of values, where p can equal to 8 and 10, 16,.. and q can equal 0, 8, 10,… and more, but to avoid overfitting our dataset and for computational sake we will consider the first three values of p and q.
Therefore, here are our possibilities for our arima dataset for AMAZON: Sarima(p = 8 or 9, d = 0, q = 0, 8, or 9, P = 1.0 or 2.0, D = 0, Q = 1)
After fine tuning our data to get the best 3 AIC values, we ended up with the three best models: ARIMA(8,0,0)(1,0,1) ARIMA(8,0,0)(2,0,1) ARIMA(9,0,0)(1,0,1)
## amzn_model1.aic amzn_model2.aic amzn_model3.aic
## 1 1171.279 1171.458 1197.487Now, we will test the residuals of our values to see if our models are reliable:
Testing models’ residuals
Our residuals should look like a normal white noise, and we can test this using a histogram.
# Test for googl:
par(mfrow=c(2,2))
hist(googl_model1$residuals, breaks = 30)
hist(googl_model2$residuals, breaks = 30)
hist(googl_model3$residuals, breaks = 30)
par(mfrow=c(2,2))
hist(nvda_model1$residuals, breaks = 30)
hist(nvda_model2$residuals, breaks = 30)
hist(nvda_model3$residuals, breaks = 30)
par(mfrow = c(2,2))
hist(msft_model1$residuals, breaks = 30)
hist(msft_model2$residuals, breaks = 30)
hist(msft_model3$residuals, breaks = 30)
par(mfrow = c(2,2))
hist(amzn_model1$residuals, breaks = 30)
hist(amzn_model2$residuals, breaks = 30)
hist(amzn_model3$residuals, breaks = 30)
Our histograms look normal and resembles a gaussian white noise distribution with the exception of NVIDIA, which is unaffected by our parameters set in our arima model. We can do another test by checking our acf and pacf graphs for all instances.
LJUNG-BOX TEST and BOX-PIERCE TESTS
## Box-Pierce Ljung-Box
## g1 0.0005869036 0.0003764588
## g2 0.0006234068 0.0004008371
## g3 0.0004514679 0.0002851218
## n1 0.9351171564 0.9173856761
## n2 0.0043621764 0.0025660295
## n3 0.8255629600 0.7858714386
## m1 0.9910585886 0.9880232353
## m2 0.9867414970 0.9816699514
## m3 0.9854728912 0.9800685009
## a1 0.4255196081 0.3743265480
## a2 0.4610717663 0.4092679633
## a3 0.1972082348 0.1601157779These are our box pierce test values from our dataset.
PACFS AND ACF OF GOOGLE MODEL RESIDUALS:
PACFS AND ACF OF NVIDIA MODEL RESIDUALS:
PACFS AND ACF OF MICROSOFT MODEL RESIDUALS:
PACFS AND ACF OF AMAZON MODEL RESIDUALS:
All in all, our pacf and acf charts for all of our stocks do not resemble a white noise distribution, and none of them pass our ljung box tests, no matter how we fine tune our models, which could be a result of our transformation that still had some trending/seasonality. This tells us that our models may not be as effective on our test data, but we can check by visualizing how our models perform on the test data of 2019.
Forecasting:
Google Forecast:
## $pred
## Time Series:
## Start = c(2019, 3)
## End = c(2020, 2)
## Frequency = 52
## [1] 53.47938 53.37517 54.15219 55.16392 53.82021 52.74956 53.73407 54.37975
## [9] 53.82974 54.81265 55.45614 53.49241 53.17454 53.07353 53.26347 53.95230
## [17] 53.94936 54.04051 55.24205 54.72484 55.71192 55.98895 56.33878 55.96124
## [25] 56.14132 55.34441 55.39723 57.78714 58.06633 58.83775 58.22608 58.24736
## [33] 57.76330 57.83559 58.32471 57.39663 57.59665 57.76780 57.98804 58.01464
## [41] 57.89426 58.18244 57.96755 58.08230 58.07986 58.07686 57.76889 58.00002
## [49] 58.20314 58.48827 58.26054 58.31127
##
## $se
## Time Series:
## Start = c(2019, 3)
## End = c(2020, 2)
## Frequency = 52
## [1] 1.152379 1.606568 1.939969 2.208894 2.435575 2.631628 2.804082 2.957604
## [9] 3.095494 3.220198 3.333593 3.437165 3.532112 3.619420 3.699916 3.774298
## [17] 3.843165 3.907035 3.966359 4.021535 4.072914 4.120807 4.165494 4.207225
## [25] 4.246225 4.282701 4.316835 4.348799 4.378745 4.406815 4.433138 4.457834
## [33] 4.481010 4.502770 4.523205 4.542403 4.560443 4.577398 4.593339 4.608329
## [41] 4.622428 4.635691 4.648170 4.659912 4.670963 4.681366 4.691158 4.700378
## [49] 4.709059 4.717233 4.724930 4.732202## $pred
## Time Series:
## Start = c(2019, 3)
## End = c(2020, 2)
## Frequency = 52
## [1] 53.26561 53.16526 53.94210 54.95918 53.60382 52.54921 53.52694 54.17091
## [9] 53.62919 54.60891 55.25374 53.29412 52.98187 52.88417 53.07287 53.75711
## [17] 53.78388 53.89040 55.07896 54.56216 55.56459 55.83579 56.18809 55.78220
## [25] 55.96148 55.15282 55.19494 57.59223 57.87853 58.62078 58.01455 58.02530
## [33] 57.54423 57.61849 58.10658 57.18409 57.37265 57.54408 57.76869 57.79956
## [41] 57.69704 57.98271 57.78360 57.89019 57.88669 57.88882 57.58113 57.78555
## [49] 58.00390 58.29417 58.06864 58.10545
##
## $se
## Time Series:
## Start = c(2019, 3)
## End = c(2020, 2)
## Frequency = 52
## [1] 1.143674 1.490121 1.760142 1.985507 2.180402 2.352675 2.507229 2.647384
## [9] 2.775519 2.893415 3.002446 3.103703 3.198072 3.286282 3.368947 3.446586
## [17] 3.519646 3.588514 3.653528 3.714987 3.773155 3.828268 3.880539 3.930159
## [25] 3.977300 4.022120 4.064762 4.105359 4.144030 4.180887 4.216033 4.249562
## [33] 4.281563 4.312118 4.341303 4.369189 4.395844 4.421328 4.445701 4.469018
## [41] 4.491330 4.512685 4.533129 4.552705 4.571454 4.589414 4.606621 4.623110
## [49] 4.638914 4.654062 4.668580 4.682518## $pred
## Time Series:
## Start = c(2019, 3)
## End = c(2020, 2)
## Frequency = 52
## [1] 53.23557 53.42090 53.95436 55.26098 53.62044 52.72995 53.53440 54.40577
## [9] 53.61898 54.86247 55.32726 53.48755 52.99275 53.06789 53.12010 53.98407
## [17] 53.85879 54.10050 55.20287 54.79747 55.69598 56.11017 56.36553 56.08459
## [25] 56.16834 55.43077 55.37485 57.89556 58.09272 58.97724 58.26244 58.36338
## [33] 57.78242 57.94368 58.38141 57.47409 57.60102 57.84739 58.02600 58.09689
## [41] 57.90427 58.22434 57.94041 58.13716 58.08393 58.12477 57.75041 58.03736
## [49] 58.20746 58.53073 58.24569 58.33736
##
## $se
## Time Series:
## Start = c(2019, 3)
## End = c(2020, 2)
## Frequency = 52
## [1] 1.149876 1.581204 1.918938 2.176454 2.407038 2.595765 2.771808 2.920305
## [9] 3.061539 3.182723 3.299256 3.400368 3.498269 3.583895 3.667175 3.740460
## [17] 3.811952 3.875169 3.936967 3.991832 4.045540 4.093384 4.140263 4.182145
## [25] 4.223207 4.259986 4.296055 4.328437 4.360197 4.388769 4.416791 4.442048
## [33] 4.466814 4.489175 4.511096 4.530919 4.550345 4.567938 4.585173 4.600802
## [41] 4.616106 4.630003 4.643603 4.655968 4.668062 4.679071 4.689833 4.699639
## [49] 4.709220 4.717960 4.726493 4.734296
Nvidia Forecast:
## $pred
## Time Series:
## Start = c(2019, 3)
## End = c(2020, 2)
## Frequency = 52
## [1] 34.00326 34.87516 35.52663 36.71819 36.92392 37.30477 37.48586 36.97857
## [9] 35.46850 36.99886 38.02906 37.51497 36.22378 34.74699 36.48814 35.94003
## [17] 36.24231 36.31519 39.74366 40.17781 40.18123 41.11585 42.79477 42.67621
## [25] 42.27218 40.28134 40.11661 42.07556 42.35994 42.59027 42.64312 42.44459
## [33] 42.75681 44.31280 45.54405 44.55376 45.55648 45.22060 45.85839 46.49429
## [41] 43.98671 43.97835 39.80596 42.18052 41.39698 38.08892 36.31603 36.65026
## [49] 36.71339 35.52816 35.01489 35.00812
##
## $se
## Time Series:
## Start = c(2019, 3)
## End = c(2020, 2)
## Frequency = 52
## [1] 1.483838 2.138545 2.629699 3.037760 3.392895 3.710382 3.999242 4.265336
## [9] 4.512732 4.744392 4.962554 5.168960 5.364998 5.551797 5.730286 5.901248
## [17] 6.065346 6.223146 6.375141 6.521760 6.663379 6.800334 6.932922 7.061409
## [25] 7.186034 7.307016 7.424549 7.538814 7.649976 7.758183 7.863577 7.966283
## [33] 8.066421 8.164101 8.259426 8.352490 8.443383 8.532188 8.618985 8.703846
## [41] 8.786840 8.868034 8.947489 9.025262 9.101411 9.175985 9.249037 9.320611
## [49] 9.390755 9.459510 9.526919 9.593033## $pred
## Time Series:
## Start = c(2019, 3)
## End = c(2020, 2)
## Frequency = 52
## [1] 34.10300 34.98563 35.63731 36.83514 37.03031 37.40665 37.60955 37.11043
## [9] 35.57461 37.12165 38.15189 37.62035 36.30922 34.82649 36.59706 36.03058
## [17] 36.32596 36.40370 39.82356 40.23671 40.23391 41.16501 42.82326 42.72416
## [25] 42.30303 40.30719 40.13925 42.08775 42.36262 42.60402 42.63939 42.45224
## [33] 42.75449 44.32410 45.56099 44.56548 45.56498 45.20606 45.85020 46.49110
## [41] 43.92304 43.90156 39.67206 42.05954 41.24091 37.87388 36.07174 36.44875
## [49] 36.51383 35.31803 34.77817 34.75785
##
## $se
## Time Series:
## Start = c(2019, 3)
## End = c(2020, 2)
## Frequency = 52
## [1] 1.485474 2.094639 2.557916 2.945029 3.283077 3.586002 3.862116 4.116851
## [9] 4.353993 4.576308 4.785892 4.984378 5.173069 5.353025 5.525125 5.690103
## [17] 5.848584 6.001103 6.148126 6.290055 6.427249 6.560022 6.688656 6.813401
## [25] 6.934485 7.052113 7.166470 7.277726 7.386036 7.491543 7.594377 7.694659
## [33] 7.792503 7.888011 7.981280 8.072401 8.161457 8.248529 8.333689 8.417007
## [41] 8.498550 8.578378 8.656550 8.733120 8.808142 8.881665 8.953734 9.024396
## [49] 9.093693 9.161664 9.228348 9.293800## $pred
## Time Series:
## Start = c(2019, 3)
## End = c(2020, 2)
## Frequency = 52
## [1] 33.97041 34.82825 35.48018 36.67198 36.87857 37.26002 37.44060 36.93332
## [9] 35.42458 36.95475 37.98540 37.47236 36.18227 34.70595 36.44638 35.89933
## [17] 36.20228 36.27531 39.70476 40.14014 40.14414 41.07931 42.75956 42.64049
## [25] 42.23747 40.24698 40.08266 42.04251 42.32761 42.55779 42.61164 42.41289
## [33] 42.72583 44.28167 45.51306 44.52318 45.52638 45.19169 45.82953 46.46552
## [41] 43.96051 43.95292 39.78280 42.15726 41.37533 38.06963 36.29804 36.63077
## [49] 36.69405 35.50930 34.99724 34.99122
##
## $se
## Time Series:
## Start = c(2019, 3)
## End = c(2020, 2)
## Frequency = 52
## [1] 1.483744 2.138307 2.637149 3.050636 3.410018 3.731013 4.022869 4.291576
## [9] 4.541288 4.775023 4.995060 5.203171 5.400768 5.588996 5.768802 5.940980
## [17] 6.106201 6.265043 6.418003 6.565518 6.707970 6.845698 6.979004 7.108158
## [25] 7.233402 7.354957 7.473022 7.587779 7.699395 7.808021 7.913800 8.016860
## [33] 8.117321 8.215296 8.310888 8.404194 8.495304 8.584302 8.671269 8.756279
## [41] 8.839402 8.920705 9.000249 9.078095 9.154297 9.228910 9.301983 9.373564
## [49] 9.443700 9.512433 9.579805 9.645872
Microsoft Forecast:
## Warning in stats::arima(xdata, order = c(p, d, q), seasonal = list(order = c(P,
## : possible convergence problem: optim gave code = 1## $pred
## Time Series:
## Start = c(2019, 3)
## End = c(2020, 2)
## Frequency = 52
## [1] 102.6548 102.5980 102.9118 105.2996 102.7551 101.5708 103.5140 103.2180
## [9] 103.3653 103.2570 104.3697 102.8611 102.4661 103.7534 103.4221 104.1997
## [17] 105.9772 103.6197 106.2591 104.3708 105.6412 106.8448 106.4328 106.7678
## [25] 106.0657 104.7533 106.0704 107.8679 110.0677 110.6298 109.2705 110.5557
## [33] 109.0781 109.4407 111.2656 109.3549 112.0509 111.2973 111.1502 112.1490
## [41] 110.3755 112.7531 113.2252 112.9252 114.6279 112.8346 111.8668 114.3449
## [49] 113.2739 114.9486 112.3076 112.5077
##
## $se
## Time Series:
## Start = c(2019, 3)
## End = c(2020, 2)
## Frequency = 52
## [1] 1.630120 2.107820 2.486350 2.837434 3.150595 3.385100 3.646234 3.882387
## [9] 4.130747 4.384554 4.607601 4.823248 5.001517 5.181451 5.370785 5.550614
## [17] 5.735946 5.895059 6.040606 6.191538 6.334698 6.491167 6.636434 6.765905
## [25] 6.895555 7.013587 7.142639 7.272768 7.391341 7.510118 7.614105 7.721521
## [33] 7.834834 7.941585 8.052421 8.149968 8.243607 8.342176 8.435822 8.536934
## [41] 8.630580 8.716650 8.805192 8.887301 8.976605 9.065002 9.146113 9.228866
## [49] 9.303264 9.381260 9.462256 9.537984## $pred
## Time Series:
## Start = c(2019, 3)
## End = c(2020, 2)
## Frequency = 52
## [1] 101.5054 101.4707 102.3680 104.5467 102.8286 100.4338 102.5441 102.8591
## [9] 101.7934 102.8182 103.2849 101.4267 101.7903 102.0151 102.7619 103.3232
## [17] 104.4658 102.9861 104.5405 103.3220 104.7982 105.2228 105.8541 105.2434
## [25] 104.7668 103.9812 104.4024 107.2468 108.7750 109.2046 108.5322 108.8588
## [33] 108.2903 108.3554 109.7293 108.6437 110.3847 110.3203 110.2845 110.5476
## [41] 109.6268 111.0693 111.7040 111.8418 112.7968 111.8906 110.1948 112.7366
## [49] 112.2382 112.9819 111.1075 110.9721
##
## $se
## Time Series:
## Start = c(2019, 3)
## End = c(2020, 2)
## Frequency = 52
## [1] 1.663114 2.141734 2.526280 2.876792 3.166694 3.400305 3.660240 3.910274
## [9] 4.168695 4.404022 4.629320 4.857016 5.065570 5.264074 5.457658 5.641983
## [17] 5.818826 5.990906 6.156771 6.317514 6.472450 6.622530 6.768247 6.909271
## [25] 7.046217 7.179495 7.308990 7.435023 7.557819 7.677458 7.794126 7.907937
## [33] 8.019041 8.127562 8.233588 8.337241 8.438621 8.537802 8.634878 8.729930
## [41] 8.823027 8.914241 9.003638 9.091282 9.177229 9.261536 9.344252 9.425433
## [49] 9.505123 9.583368 9.660207 9.735695## $pred
## Time Series:
## Start = c(2019, 3)
## End = c(2020, 2)
## Frequency = 52
## [1] 101.4794 101.4594 102.5017 104.5738 102.6846 100.5006 102.7297 102.6837
## [9] 101.8753 102.6801 103.3980 101.3687 101.8680 102.0226 102.8041 103.2824
## [17] 104.4999 102.9840 104.5276 103.3316 104.7775 105.2125 105.8110 105.2264
## [25] 104.7424 103.9764 104.3858 107.2167 108.7065 109.1364 108.4715 108.7971
## [33] 108.2386 108.2980 109.6449 108.5855 110.3006 110.2342 110.2019 110.4681
## [41] 109.5546 111.0074 111.6363 111.7783 112.7112 111.8207 110.1498 112.6556
## [49] 112.1640 112.9034 111.0596 110.9209
##
## $se
## Time Series:
## Start = c(2019, 3)
## End = c(2020, 2)
## Frequency = 52
## [1] 1.659443 2.139232 2.523027 2.872777 3.161786 3.394450 3.655806 3.905833
## [9] 4.163302 4.398727 4.625765 4.862186 5.063084 5.265958 5.457852 5.642381
## [17] 5.818683 5.992649 6.157878 6.319694 6.474169 6.624987 6.770963 6.911695
## [25] 7.049355 7.182654 7.312433 7.438692 7.561873 7.681621 7.798676 7.912630
## [33] 8.024095 8.132827 8.239135 8.343084 8.444731 8.544195 8.641563 8.736904
## [41] 8.830285 8.921806 9.011488 9.099443 9.185683 9.270300 9.353321 9.434811
## [49] 9.514812 9.593370 9.670522 9.746329
Amazon Forecast:
## $pred
## Time Series:
## Start = c(2019, 3)
## End = c(2020, 2)
## Frequency = 52
## [1] 75.20012 75.41663 75.78935 76.09745 76.24745 74.01561 75.49481 76.38496
## [9] 75.85374 77.02177 76.92799 76.14028 74.46018 74.89334 75.08870 77.22944
## [17] 79.21360 77.71169 79.05457 78.51039 79.35412 79.98349 80.75455 80.91559
## [25] 81.15560 80.51326 80.13052 82.53478 83.44538 83.97127 82.84576 83.15064
## [33] 82.89052 83.16151 85.10011 83.97747 84.98399 84.05166 84.85921 83.82398
## [41] 82.54323 81.87719 80.53975 81.35038 82.54776 80.65725 80.12613 82.31290
## [49] 82.72806 81.76484 78.84149 79.05355
##
## $se
## Time Series:
## Start = c(2019, 3)
## End = c(2020, 2)
## Frequency = 52
## [1] 1.511703 2.135902 2.544135 2.902332 3.269234 3.694433 4.071228
## [8] 4.454978 4.807687 5.135525 5.451087 5.750721 6.041292 6.314194
## [15] 6.574227 6.821735 7.057366 7.282901 7.497883 7.703669 7.900259
## [22] 8.088462 8.268870 8.441831 8.607901 8.767354 8.920639 9.068072
## [29] 9.209976 9.346661 9.478381 9.605404 9.727954 9.846257 9.960517
## [36] 10.070923 10.177654 10.280874 10.380742 10.477402 10.570991 10.661639
## [43] 10.749466 10.834588 10.917112 10.997140 11.074769 11.150090 11.223188
## [50] 11.294147 11.363044 11.429956## $pred
## Time Series:
## Start = c(2019, 3)
## End = c(2020, 2)
## Frequency = 52
## [1] 76.36389 79.22067 80.29135 83.57804 86.54480 81.52916 85.52739
## [8] 86.86213 85.20352 89.73194 90.34570 88.03053 82.80603 83.69383
## [15] 83.99618 89.10443 93.24872 89.71718 91.83856 90.60230 92.14107
## [22] 93.59893 95.85788 96.91224 98.45584 96.49584 95.03365 100.44835
## [29] 101.42920 103.55610 101.28162 102.84110 102.79510 103.65976 108.63097
## [36] 105.17138 107.89119 105.11688 107.28785 103.44157 98.94569 97.52056
## [43] 92.71378 94.80965 98.11099 91.38460 88.32644 96.42631 97.46065
## [50] 94.46038 86.27018 86.84764
##
## $se
## Time Series:
## Start = c(2019, 3)
## End = c(2020, 2)
## Frequency = 52
## [1] 1.781239 2.537085 3.060347 3.515287 3.966303 4.447681 4.874473
## [8] 5.301467 5.692977 6.054147 6.393263 6.710946 7.012487 7.293246
## [15] 7.557366 7.805299 8.038318 8.257936 8.464751 8.660030 8.844274
## [22] 9.018398 9.183101 9.339005 9.486733 9.626785 9.759681 9.885860
## [29] 10.005744 10.119719 10.228137 10.331327 10.429589 10.523205 10.612435
## [36] 10.697521 10.778686 10.856142 10.930083 11.000695 11.068146 11.132598
## [43] 11.194200 11.253094 11.309427 11.363299 11.414866 11.464212 11.511436
## [50] 11.556629 11.599910 11.641516## $pred
## Time Series:
## Start = c(2019, 3)
## End = c(2020, 2)
## Frequency = 52
## [1] 75.41590 76.81819 76.89253 75.60861 76.51399 75.50135 77.62838 77.59186
## [9] 76.96842 78.39002 78.62027 78.36573 76.63748 77.18112 77.45898 79.74089
## [17] 81.75692 80.36967 81.79496 81.38512 82.42260 83.07651 83.86584 84.06005
## [25] 84.35058 83.90341 83.64775 85.99473 86.98436 87.51823 86.55268 86.85106
## [33] 86.68529 87.00431 88.87728 87.97950 88.98652 88.21762 89.02222 88.25219
## [41] 87.27968 86.74400 85.76282 86.60982 87.79961 86.28504 85.99418 87.93237
## [49] 88.37930 87.62724 85.10246 85.37888
##
## $se
## Time Series:
## Start = c(2019, 3)
## End = c(2020, 2)
## Frequency = 52
## [1] 1.499110 2.137935 2.551417 2.917779 3.290698 3.732943 4.129577
## [8] 4.523588 4.832385 5.173574 5.519433 5.840567 6.147885 6.432484
## [15] 6.720476 6.995494 7.267908 7.524716 7.773155 8.018754 8.256356
## [22] 8.489544 8.713951 8.934437 9.149402 9.360383 9.566840 9.768124
## [29] 9.966132 10.159972 10.350763 10.537761 10.721582 10.902341 11.080138
## [36] 11.255332 11.427643 11.597519 11.764865 11.929944 12.092767 12.253399
## [43] 12.411985 12.568532 12.723205 12.875990 13.026998 13.176267 13.323864
## [50] 13.469855 13.614270 13.757179
Forecast Results, Findings, and Inferences:
Our Google stock had a smooth line, so our prediction assumed that we did not have a seasonality and instead a gradual, positive trend. The arima model prediction was fairly close with Google’s true value in blue until the 3rd quarter of 2019, when Google’s stock increased much more than predicted. The same applies for all of our arima models fitted on the google dataset.
Our Nvidia stock forecast did manage to imitate the true values. However, it veered off trajectory in the 3rd quarter of 2019. Nevertheless, this is our second best prediction and it performed fairly well.
Our Microsoft stock prediction were far off from what we had predicted despite its stock having a steady increase. The forecast most likely used its current stagnant trend to make a prediction, which was when the stock decided to increase in price since the beginning of 2019.
Our Amazon stock forecast was our most accurate forecast. Albeit having a stable trend, the prediction accurately determined its trajectory up until the end of 2020. Our 2nd Arima model correctly guessed the majority of Amazon’s price trajectory throughout 2019. We can see that the true value was within our forecast’s confidence bands for almost the entirety of the true values, though this is not much as these stocks are not very volatile.
Overall, we can conclude that we did not achieve the forecasts that we wanted, but if we were to forecast our stocks, we would run our arima models on the Amazon stock for its predictability. It seemed our forecasts did also did not account for the fact that we had seasonality in our dataset. It seems the higher the values of our p, q, P, and Q, the lesser the accurate we are in our models. This could be due to the fact that we have overfit on our training set and thus performed poorly on the test set. We also have shown that we need to find a problem with either our transformation or our model, because we did not get gaussian white noise distributions from our residuals after we fit the data. All in all, the forecasts and models did not fit, but some made decent predictions, especially Amazon. We can also infer from the fact that these are incredibly stable stocks and not as volatile as other riskier stocks. Thus, we have a more stable true prediction which could contribute to having a good forecast despite our evaluation of our models not being very strong.
There were other noticeable factors that could have played into our poor stock forecast as well. One was notably the US-China trade war, which began in mid-July which was towards the end of our training dataset. This event crippled the technology industry especially, with the industry being heavily reliant on labor and manufacturing parts from China.
Conclusion:
In conclusion, our analysis of the weekly stock prices for four of the largest technology companies in the world over a 10-year period from 2010 to 2020 has provided us with interesting insights into the performance of these stocks. Our main objective was to determine which of the four stocks would have the highest predicted performance in the year 2019, and thus we used ARIMA models to make predictions.
After analyzing the data and comparing our predictions to the true values, we found that our models did not perform as accurately as we had hoped. Our Google stock had a smooth line, and our prediction assumed that we did not have seasonality, but our ARIMA model was fairly close until the 3rd quarter of 2019, when Google’s stock increased more than predicted. Our Nvidia stock forecast managed to imitate the true values until it veered off course towards the 3rd quarter of 2019 as well, but our Microsoft stock prediction was far off from what we had predicted despite its stock having a steady increase. Finally, our Amazon stock forecast was our most accurate forecast, with the second ARIMA model making correct predictions of the stock’s price trajectory throughout 2019.
We can conclude that our ARIMA models did not perform as accurately as we had hoped, but our analysis did provide us with some valuable insights. For instance, we observed that Amazon was the most predictable stock among the four, and thus it could be an excellent choice for investors who wish to invest in individual stocks from the technology industry.
Overall, our analysis highlighted the importance of using statistical models to make stock predictions, especially in industries as dynamic as the technology sector. While our models did not provide us with the accuracy we were hoping for, it is still a valuable exercise in understanding the nuances of the stock market and making informed investment decisions. Future analyses could include more factors and consider the impact of major events on the stocks, such as pandemics, economic crises, and geopolitical tensions.
Appendix:
Datasets retrieved from Yahoo Finance:
Amazon. (n.d.). Retrieved from https://finance.yahoo.com/quote/AMZN/history?p=AMZN
Microsoft. (n.d.). Retrieved from https://finance.yahoo.com/quote/MSFT/history?p=MSFT
Nvidia. (n.d.). Retrieved from https://finance.yahoo.com/quote/NVDA/history?p=NVDA
Google. (n.d.). Retrieved from https://finance.yahoo.com/quote/GOOGL/history?p=GOOGL